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Coupling Into Waveguide Evanescent Modes with Applications in Electron Paramagnetic Resonance Jason Walter Sidabras Marquette University

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Coupling Into Waveguide Evanescent Modes with Applications in Electron Paramagnetic Resonance Jason Walter Sidabras Marquette University Marquette University e-Publications@Marquette Master's Theses (2009 -) Dissertations, Theses, and Professional Projects Coupling into Waveguide Evanescent Modes with Applications in Electron Paramagnetic Resonance Jason Walter Sidabras Marquette University Recommended Citation Sidabras, Jason Walter, "Coupling into Waveguide Evanescent Modes with Applications in Electron Paramagnetic Resonance" (2010). Master's Theses (2009 -). Paper 29. http://epublications.marquette.edu/theses_open/29 COUPLING INTO WAVEGUIDE EVANESCENT MODES WITH APPLICATIONS IN ELECTRON PARAMAGNETIC RESONANCE by Jason W. Sidabras A Thesis submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree of Master of Science. Milwaukee, Wisconsin May 2010 ABSTRACT COUPLING INTO WAVEGUIDE EVANESCENT MODES WITH APPLICATIONS IN ELECTRON PARAMAGNETIC RESONANCE Jason W. Sidabras, B.S. Marquette University, 2010 The use of analytical and numerical techniques in solving the coupling of evanescent modes in a microwave waveguide through slots can be optimized to create a uniform magnetic field excitation on axis within a waveguide. This work has direct applications in Electron Paramagnetic Resonance (EPR) where a 100 kHz time-varying magnetic field is incident on a sample contained in a microwave cavity. Typical cavity designs do not take into consideration the uniformity of the 100 kHz field modulation and assume it to be uniform enough over the sample region from quasi-static principles. This work shows otherwise and uses Ansoft (Pittsburgh, PA) High Frequency Structure Simulator (HFSS; version 12.0) and analytical dyadic Green's functions to understand the coupling mechanisms. The techniques described in this work have shown that electromagnetic modes form in a rectangular and cylindrical waveguide domain even at frequencies a number of orders of magnitude below the waveguide cut-off frequencies. With slot thicknesses very small compared to a wavelength, Born's first approximation must be modified to account for a near field secondary wave. Additionally, mutual coupling between multiple slots has been shown to influence the overall magnetic field profile down the axis of the waveguide and in certain circumstances becomes more complex from interactions outside of the domain of the dyadic Green's functions. A cylindrical TE01U cavity resonant at W-band (94 GHz) is proposed where both the microwave magnetic field and, from this work, the 100 kHz time-varying magnetic field incident on the sample are uniform. This type of resonator is highly desirable in EPR experiments where inhomogeneity of magnetic fields affect signal purity. With the technology outlined in this work, experiments where a uniform field modulation amplitude is swept over the entire spectra to obtain pure absorption is feasible. This work advances the cutting edge of resonator design and enables new experiments to be performed at high field EPR. i ACKNOWLEDGEMENTS This work as supported by grants EB001417 and EB001980 from the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health. I would like to thank Dr. James S. Hyde for the opportunity as an engineer at the Medical College of Wisconsin and the support for furthering my education at Marquette University. I would also like to thank Dr. James E. Richie for the numerous hours of discussion, guidance, and support throughout this work and Dr. Richard R. Mett for continuing to push me towards bettering myself. Finally, I would like to thank my family and especially my girlfriend Emily for the support and needed push to finish this work. I couldn't have done it without all of you. ii TABLE OF CONTENTS 1 Introduction 1 2 Theory 5 2.1 Electron Paramagnetic Resonance . 5 2.1.1 Field Modulation Coupling Techniques . 12 2.1.2 Proposed Uniform Field Modulation Cavity Design . 14 2.2 Electromagnetic Solutions . 15 2.2.1 Vector Potentials . 16 2.2.2 Boundary Conditions . 18 2.2.3 Eigenfunctions and Eigenvalues . 19 2.2.4 Green's Functions . 22 2.2.5 Dyadic Green's Functions . 26 2.2.6 Rectangular Waveguide Formulation . 29 2.2.7 Cylindrical Waveguide Formulation . 34 2.2.8 Equivalence Principle . 36 2.3 Computational Methods . 42 2.3.1 Finite-Element Method . 43 2.4 Methods . 46 2.4.1 Results and Discussions . 47 3 Results 49 3.1 Single Slot Eddy Current Analysis . 49 3.2 Integration of the Source . 53 3.3 Single Slot Rectangular Results . 55 3.4 Single Slot Cylindrical Results . 62 4 Discussion 67 4.1 Multiple Slot Formulation . 67 4.1.1 Summation of Multiple Slot Fields . 67 4.1.2 Moment Method . 71 iii 5 Conclusion 77 5.1 Recommendations and Future Work . 78 Bibliography 79 APPENDIX - Dyadic Mathematical Properties 84 iv LIST OF FIGURES 1.1 Important applied fields in an EPR experiment . 1 2.1 Cylindrical TE011 Cavity. 7 2.2 Basic EPR bridge setup . 8 2.3 Description of field modulation on an EPR absorption spectra . 10 2.4 Calculated spectra of varying modulation field amplitudes . 11 2.5 Illustration of field modulation techniques . 13 2.6 Cylindrical TE01U optimized for uniform fields . 15 2.7 Methods for computing electromagnetic fields . 16 2.8 Definition of the rectangular geometry . 29 2.9 Definition of the cylindrical geometry . 34 2.10 The equivalence principle. 37 2.11 Deviation of B around a PEC waveguide . 38 2.12 Formulation of slot sources using equivalence principle . 39 2.13 Formulation of the Magnetic Current using Ezz^ . 40 2.14 Wavelength variations of Born's first approximation . 41 2.15 Characterization of magnetic surface current source . 42 2.16 Arbitrary finite-element method domain . 44 2.17 Maxwell 3D and HFSS surface current comparison . 47 3.1 Numerical eddy-current plot of J e on a finite sized PEC plate using Ansoft HFSS . 50 3.2 Analytical eddy-current plot of calculated J e on a finite size plate . 51 3.3 Characterization of magnetic surface current source filament and convolution 54 3.4 Rectangular geometry used for a single slot . 56 3.5 Rectangular Waveguide Solutions using dyadic Green's functions . 57 3.6 Rectangular Waveguide Solutions using Ansoft HFSS . 57 3.7 Magnetic field profile for the rectangular waveguidez ^-axis . 58 3.8 Variations in magnetic field on axis versus slot depth . 59 v 3.9 Variations in magnetic current profile around the slot . 60 3.10 Numerical and analytical phase calculations . 61 3.11 Definition of the cylindrical geometry . 63 3.12 Variations in a magnetic current profile around a cylindrical slot . 63 3.13 Cylindrical Waveguide Solutions using dyadic Green's Functions . 64 3.14 Cylindrical Waveguide Solutions using Ansoft HFSS . 65 3.15 Magnetic field profile for the cylindrical waveguidez ^-axis . 66 4.1 Multiple slot configurations using a simple summation of the slot fields . 68 4.2 Mutual coupling of two slots. 71 4.3 Visualization of the moment method. 72 4.4 Multiple slot configurations using first-order moment method . 74 4.5 Outside waveguide domain interactions . 76 vi LIST OF TABLES 2.1 Helmholtz wave equation solution set . 20 3.1 Phase between excitation coil and domain . 62 4.1 RMSE calculations for a simple summation of multiple slot fields . 69 4.2 RMSE calculations for first-order moment method multiple slot formulation 75 1 Chapter 1: Introduction In an Electron Paramagnetic Resonance (EPR) experiment, there are three fields incident on a differential sample volume to excite and measure magnetic resonance: i) a uniform static magnetic field from a super-conducting magnet, ii) the RF field in a microwave cavity, and iii) modulated field from an external coil added to the static magnetic field [1, 2, 3]. An illustration of these fields is found in Fig. 1.1. Here, the static and RF field, H0 and H1 respectively, are spatially perpendicular while a time-varying field is added parallel to the static magnetic field. Figure 1.1: There are three fields that are applied to a sample (solid blue) in an EPR experiment: i) the static magnetic field (purple), ii) the RF microwave field applied spatially perpendicular to the static magnetic field (dashed), and iii) the applied field modulation applied spatially parallel to the static magnetic field (green). Over the last five years research has been conducted to improve the uniformity of the RF magnetic field in a microwave cavity, where typically a cavity has a cosine dependence [4, 5, 6]. Using a waveguide section at cut-off over the region of interest and proper end-sections to tune the cavity to the cut-off frequency, a purely uniform field can be realized. The cavity is then immersed in a static magnetic field whose amplitude is swept slowly over the resonance condition. These recent advancements of resonator technology using a waveguide section at cut-off achieve very good RF uniformity, yet there has been no literature to the author's knowledge about creating uniform field modulation by the techniques discussed in this work. Past and current literature assumes that the 100 kHz field modulation has such a large wavelength compared to the resonator body that it can be regarded as quasi-static [1, 7]. A quasi-static field is defined as a field which has no wave-like properties, such that 2 @B=@t = 0. The penetration of field modulation into a resonator is described as \good enough" and is mostly assumed to be uniform in the waveguide cross-section and cosine down the axis of the sample. This work shows otherwise. The use of commercially available finite-element modeling software, Ansoft (Pittsburgh, PA) High Frequency Structure Simulator (HFSS; version 12.0), provides an introduction to the understanding of the coupling of 100 kHz field modulation into a rectangular and cylindrical waveguide with slots cut perpendicular to the axis.
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